Deeply rattled by President Trump’s insistence that he wants to acquire Greenland, Prime Minister Mette Frederiksen of Denmark met European and NATO leaders on a whirlwind tour on Tuesday, traveling to Berlin, Paris and Brussels to garner public support and discuss how best to respond to the sudden demands of one of Denmark’s closest allies.

The Danes are seeking allied solidarity while at the same time urging their European partners in both the European Union and NATO not to inflame the situation and make matters worse before Mr. Trump’s strategy becomes clear, if he has formulated one.

“Frederiksen doesn’t want to get into a one-on-one fight with Donald Trump, because she will lose,” said Ulrik Pram Grad of the Danish Institute of International Studies. “She’s trying to get colleagues to commit to the fact that this is not just her problem, but a problem for Europe, if tariffs are imposed” — referring to Mr. Trump’s threat of tariffs if he does not get his way — “and a problem for NATO, if an ally threatens to take away sovereign territory of another.”

On Monday, just before Ms. Frederiksen set out on her journey, her government announced that it would increase military spending in the North Atlantic by the equivalent of $2 billion. The new spending has been planned for some time, but its announcement appeared to have been moved up in response to Mr. Trump, Mr. Grad said.

Just days into his second term, Mr. Trump said he wanted the United States to take control of Greenland as “an absolute necessity” for Western security and refused to rule out using military or economic force to do so.

If Mr. Trump uses tariffs to pressure Denmark, the European Union could respond with countertariffs or even use a special “anti-coercion instrument” adopted in 2023, which offers a range of countermeasures. But those are meant as a deterrent, and Denmark and the Europeans would much prefer some sort of deal with Mr. Trump — though one short of surrendering sovereignty. Many European officials are already comparing Mr. Trump’s demands to how President Vladimir V. Putin of Russia has insisted that Ukraine surrender Crimea and four other regions.

The European Union is itself divided, said Fabian Zuleeg, chief executive of the European Policy Centre in Brussels. “There will be some countries that will support Trump no matter what, given their ideologies and a bit of glee about how liberal democracies are under pressure,” he said. The Danish prime minister’s trip was more about “shoring up support among like-minded countries” and inside NATO itself.

So far Mr. Trump’s threats are only words, Mr. Zuleeg noted. “But Europe is vulnerable, because the ultimate threat over our heads is the withdrawal of the U.S. security guarantee,” which is more important than the interests of one country, he said. “So I suspect there will have to be an effort to give the U.S. something, some of the economic interests behind this extreme request, but not so far as to hand Greenland over to America.”

As for NATO, he said, “this will undermine the alliance, because it is simply not an acceptable way of interacting with allies and it will poison the well on other matters.”

But Washington remains a dominant actor. “How do you deter the United States of America, your chief security provider?” asked Leslie Vinjamuri, who heads the U.S. and Americas Program at Chatham House in London.

Ms. Frederiksen and her government have already offered the United States the chance to work together to enhance security around Greenland. The new spending announced on Monday includes three new Arctic naval vessels to patrol waters around Greenland and the Faroe Islands, and two more long-range drones and satellites to improve surveillance of the area, according to the government’s statement announcing the spending. There will also be training for young Greenlanders “to take responsibility for preparedness,” the statement said.

“We must face the fact that there are serious challenges regarding security and defense in the Arctic and North Atlantic,” Troels Lund Poulsen, Denmark’s defense minister, said in the statement.

But under existing agreements from 1951, as Mr. Grad of the Danish Institute pointed out, the United States has “almost full military sovereignty over Greenland,” where it already has a large military base. All Washington needs to do to take some kind of action is to consult with Denmark and Greenland, which would have no reason to object, he said.

On Saturday, the chairman of the European Union Military Committee, Gen. Robert Brieger, told the German newspaper Welt am Sonntag that “it would make perfect sense not only to station U.S. forces in Greenland, as is currently the case, but also to consider stationing E.U. soldiers there.”

For NATO, there is a legitimate strategic issue about Greenland for allies to discuss, said Camille Grand, a former senior NATO official now with the European Council on Foreign Relations.

“But the Danes are America’s best allies, who suffered serious casualties in Iraq and Afghanistan fighting alongside the United States, and they buy American,” he said. “It’s hard to find a more trans-Atlanticist country in Europe. So why bully the Danes? It looks like Crimea. You will say that Putin’s behavior is unacceptable and people will say, ‘What about Greenland?’”

Greenland, a gigantic island in the North Atlantic that is an semiautonomous territory of Denmark, has taken on strategic importance in recent years as Arctic ice melts, opening up shipping lanes for international business, warships and submarines. With some 60,000 inhabitants, Greenland has home rule, but Denmark is responsible for its foreign and defense policy.

Greenland has a strong independence movement that is likely to take heart from Mr. Trump’s challenge to Denmark, but there is no indication that Greenlanders would then wish to hand over their independence to Washington. But an independent Greenland would very likely want a strong defense and economic relationship with the United States.

“As a government our job is not to panic and figure out what the actual demands are,” Naaja Nathanielsen, Greenland’s trade and justice minister, told Agence France-Presse. “If it is about military presence, the U.S. has been here for 80 years; we are not opposed to that. If it is about the minerals, it is an open market,” she said, referring to the plentiful natural resources on Greenland.

But “if it is about expansionism,” she said, “we are a democracy, we are allies, and we ask our allies to respect our institutions.”

Over the weekend, Ms. Frederiksen met with her Nordic allies — the leaders of Finland, Sweden, Norway. All, she said, “shared the gravity of the situation.”

In Berlin on Tuesday, after his meeting with Ms. Frederiksen, Chancellor Olaf Scholz of Germany spoke about Russia’s invasion of Ukraine and said in German that “borders must not be moved by force,” then said in English: “To whom it may concern.” He emphasized that “Denmark and Germany are close friends.”

In Paris, Ms. Frederiksen met President Emmanuel Macron of France and later headed to Brussels to meet Mark Rutte, the secretary general of NATO.

“The clear message from friends in the Nordic countries and Europe, and also outside Europe, is that there must of course be respect for territories and the sovereignty of states,” Ms. Frederiksen told Danish broadcaster TV2. “This is crucial for the international community we have built together since the Second World War.”

Her foreign minister, Lars Lokke Rasmussen, had tougher words on Tuesday.

“Trump will not have Greenland,” he said. “Greenland is Greenland. And the Greenlandic people are a people, also in the sense of international law.” He added, “This is also why we have said time and again that it is ultimately Greenland that decides Greenland’s situation.”

Maya Tekeli contributed reporting from Copenhagen.